Yes, if you understand the tensor product topology in the right way. 
Since the spaces are nuclear, inductive and projective tensor products coincide.
The result is theorem 51.6 of Treves: Topological vector spaces, distributions, and kernels.

# Added later:
Attention: The description of  seminorms on the tensor-product given in the question is not sufficient to specify a locally convex topology on the tensor product. There are many satisfying this description; all between the the projective one and the the inductive one and even more. See the source I have given, or many other books.